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Supervised Learning : Exploring Classification Models for Predicting Categorical Responses

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John Pauline Pineda

November 29, 2023

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• 1. Table of Contents
  • 1.1 Data Background
  • 1.2 Data Description
  • 1.3 Data Quality Assessment
  • 1.4 Data Preprocessing
    • 1.4.1 Data Cleaning
    • 1.4.2 Missing Data Imputation
    • 1.4.3 Outlier Treatment
    • 1.4.4 Collinearity
    • 1.4.5 Shape Transformation
    • 1.4.6 Centering and Scaling
    • 1.4.7 Data Encoding
    • 1.4.8 Preprocessed Data Description
  • 1.5 Data Exploration
    • 1.5.1 Exploratory Data Analysis
    • 1.5.2 Hypothesis Testing
  • 1.6 Model Development With Hyperparameter Tuning
    • 1.6.1 Premodelling Data Description
    • 1.6.2 Logistic Regression
    • 1.6.3 Decision Trees
    • 1.6.4 Random Forest
    • 1.6.5 Support Vector Machine
  • 1.7 Model Development With Class Weights
    • 1.7.1 Premodelling Data Description
    • 1.7.2 Logistic Regression
    • 1.7.3 Decision Trees
    • 1.7.4 Random Forest
    • 1.7.5 Support Vector Machine
  • 1.8 Model Development With SMOTE Upsampling
    • 1.8.1 Premodelling Data Description
    • 1.8.2 Logistic Regression
    • 1.8.3 Decision Trees
    • 1.8.4 Random Forest
    • 1.8.5 Support Vector Machine
  • 1.9 Model Development With CNN Downsampling
    • 1.9.1 Premodelling Data Description
    • 1.9.2 Logistic Regression
    • 1.9.3 Decision Trees
    • 1.9.4 Random Forest
    • 1.9.5 Support Vector Machine
  • 1.10 Model Development With Stacking Ensemble Learning
    • 1.9.1 Premodelling Data Description
    • 1.9.2 Logistic Regression
  • 1.11 Consolidated Findings
• 2. Summary
• 3. References

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1. Table of Contents

This project implements different predictive modelling procedures for dichotomous categorical responses using various helpful packages in Python. Models applied in the analysis to predict dichotomous categorical responses included the Logistic Regression, Decision Trees, Random Forest, Naive Bayes and Support Vector Machine algorithms. Remedial procedures on addressing class imbalance including Class Weighting, Synthetic Minority Oversampling Technique and Condensed Nearest Neighbors were similarly considered, as applicable. Ensemble learning using Stacking which consolidate many different models types on the same data and using another model to learn how to best combine the predictions was also explored. All results were consolidated in a Summary presented at the end of the document.

Binary classification learning refers to a predictive modelling problem where only two class labels are predicted for a given sample of input data. These models use the training data set and calculate how to best map instances of input data to the specific class labels. Typically, binary classification tasks involve one class that is the normal state (assigned the class label 0) and another class that is the abnormal state (assigned the class label 1). It is common to structure a binary classification task with a model that predicts a Bernoulli probability distribution for each instance. The Bernoulli distribution is a discrete probability distribution that covers a case where an event will have a binary outcome as either a 0 or 1. For a binary classification, this means that the model predicts a probability of an instance belonging to class 1, or the abnormal state. The algorithms applied in this study attempt to categorize the input data and form dichotomous groups based on their similarities.

1.1. Data Background

Datasets used for the analysis were separately gathered and consolidated from various sources including:

1. Cancer Rates from World Population Review
2. Social Protection and Labor Indicator from World Bank
3. Education Indicator from World Bank
4. Economy and Growth Indicator from World Bank
5. Environment Indicator from World Bank
6. Climate Change Indicator from World Bank
7. Agricultural and Rural Development Indicator from World Bank
8. Social Development Indicator from World Bank
9. Health Indicator from World Bank
10. Science and Technology Indicator from World Bank
11. Urban Development Indicator from World Bank
12. Human Development Indices from Human Development Reports
13. Environmental Performance Indices from Yale Center for Environmental Law and Policy

This study hypothesized that various global development indicators and indices influence cancer rates across countries.

The target variable for the study is:

• CANRAT - Dichotomized category based on age-standardized cancer rates, per 100K population (2022)

The predictor variables for the study are:

• GDPPER - GDP per person employed, current US Dollars (2020)
• URBPOP - Urban population, % of total population (2020)
• PATRES - Patent applications by residents, total count (2020)
• RNDGDP - Research and development expenditure, % of GDP (2020)
• POPGRO - Population growth, annual % (2020)
• LIFEXP - Life expectancy at birth, total in years (2020)
• TUBINC - Incidence of tuberculosis, per 100K population (2020)
• DTHCMD - Cause of death by communicable diseases and maternal, prenatal and nutrition conditions, % of total (2019)
• AGRLND - Agricultural land, % of land area (2020)
• GHGEMI - Total greenhouse gas emissions, kt of CO2 equivalent (2020)
• RELOUT - Renewable electricity output, % of total electricity output (2015)
• METEMI - Methane emissions, kt of CO2 equivalent (2020)
• FORARE - Forest area, % of land area (2020)
• CO2EMI - CO2 emissions, metric tons per capita (2020)
• PM2EXP - PM2.5 air pollution, population exposed to levels exceeding WHO guideline value, % of total (2017)
• POPDEN - Population density, people per sq. km of land area (2020)
• GDPCAP - GDP per capita, current US Dollars (2020)
• ENRTER - Tertiary school enrollment, % gross (2020)
• HDICAT - Human development index, ordered category (2020)
• EPISCO - Environment performance index , score (2022)

1.2. Data Description

1. The dataset is comprised of:
   • 177 rows (observations)
   • 22 columns (variables)
     • 1/22 metadata (object)
       • COUNTRY
     • 1/22 target (categorical)
       • CANRAT
     • 19/22 predictor (numeric)
       • GDPPER
       • URBPOP
       • PATRES
       • RNDGDP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • RELOUT
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • ENRTER
       • EPISCO
     • 1/22 predictor (categorical)
       • HDICAT

##################################
# Loading Python Libraries
##################################
import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import itertools
%matplotlib inline

from operator import add,mul,truediv
from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PowerTransformer
from sklearn.preprocessing import StandardScaler
from scipy import stats

from sklearn.linear_model import LogisticRegression
from sklearn.tree import DecisionTreeClassifier
from sklearn.ensemble import RandomForestClassifier
from sklearn.svm import SVC
from sklearn.metrics import accuracy_score,precision_score,recall_score,f1_score,roc_auc_score
from sklearn.model_selection import train_test_split, GridSearchCV

##################################
# Loading the dataset
##################################
cancer_rate = pd.read_csv('CategoricalCancerRates.csv')

##################################
# Performing a general exploration of the dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate.dtypes)

Column Names and Data Types:

COUNTRY     object
CANRAT      object
GDPPER     float64
URBPOP     float64
PATRES     float64
RNDGDP     float64
POPGRO     float64
LIFEXP     float64
TUBINC     float64
DTHCMD     float64
AGRLND     float64
GHGEMI     float64
RELOUT     float64
METEMI     float64
FORARE     float64
CO2EMI     float64
PM2EXP     float64
POPDEN     float64
ENRTER     float64
GDPCAP     float64
HDICAT      object
EPISCO     float64
dtype: object

##################################
# Taking a snapshot of the dataset
##################################
cancer_rate.head()

	COUNTRY	CANRAT	GDPPER	URBPOP	PATRES	RNDGDP	POPGRO	LIFEXP	TUBINC	DTHCMD	...	RELOUT	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	ENRTER	GDPCAP	HDICAT	EPISCO
0	Australia	High	98380.63601	86.241	2368.0	NaN	1.235701	83.200000	7.2	4.941054	...	13.637841	131484.763200	17.421315	14.772658	24.893584	3.335312	110.139221	51722.06900	VH	60.1
1	New Zealand	High	77541.76438	86.699	348.0	NaN	2.204789	82.256098	7.2	4.354730	...	80.081439	32241.937000	37.570126	6.160799	NaN	19.331586	75.734833	41760.59478	VH	56.7
2	Ireland	High	198405.87500	63.653	75.0	1.23244	1.029111	82.556098	5.3	5.684596	...	27.965408	15252.824630	11.351720	6.768228	0.274092	72.367281	74.680313	85420.19086	VH	57.4
3	United States	High	130941.63690	82.664	269586.0	3.42287	0.964348	76.980488	2.3	5.302060	...	13.228593	748241.402900	33.866926	13.032828	3.343170	36.240985	87.567657	63528.63430	VH	51.1
4	Denmark	High	113300.60110	88.116	1261.0	2.96873	0.291641	81.602439	4.1	6.826140	...	65.505925	7778.773921	15.711000	4.691237	56.914456	145.785100	82.664330	60915.42440	VH	77.9

5 rows × 22 columns

##################################
# Setting the levels of the categorical variables
##################################
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].astype('category')
cancer_rate['CANRAT'] = cancer_rate['CANRAT'].cat.set_categories(['Low', 'High'], ordered=True)
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].astype('category')
cancer_rate['HDICAT'] = cancer_rate['HDICAT'].cat.set_categories(['L', 'M', 'H', 'VH'], ordered=True)

##################################
# Performing a general exploration of the numeric variables
##################################
print('Numeric Variable Summary:')
display(cancer_rate.describe(include='number').transpose())

Numeric Variable Summary:

	count	mean	std	min	25%	50%	75%	max
GDPPER	165.0	45284.424283	3.941794e+04	1718.804896	13545.254510	34024.900890	66778.416050	2.346469e+05
URBPOP	174.0	59.788121	2.280640e+01	13.345000	42.432750	61.701500	79.186500	1.000000e+02
PATRES	108.0	20607.388889	1.340683e+05	1.000000	35.250000	244.500000	1297.750000	1.344817e+06
RNDGDP	74.0	1.197474	1.189956e+00	0.039770	0.256372	0.873660	1.608842	5.354510e+00
POPGRO	174.0	1.127028	1.197718e+00	-2.079337	0.236900	1.179959	2.031154	3.727101e+00
LIFEXP	174.0	71.746113	7.606209e+00	52.777000	65.907500	72.464610	77.523500	8.456000e+01
TUBINC	174.0	105.005862	1.367229e+02	0.770000	12.000000	44.500000	147.750000	5.920000e+02
DTHCMD	170.0	21.260521	1.927333e+01	1.283611	6.078009	12.456279	36.980457	6.520789e+01
AGRLND	174.0	38.793456	2.171551e+01	0.512821	20.130276	40.386649	54.013754	8.084112e+01
GHGEMI	170.0	259582.709895	1.118550e+06	179.725150	12527.487367	41009.275980	116482.578575	1.294287e+07
RELOUT	153.0	39.760036	3.191492e+01	0.000296	10.582691	32.381668	63.011450	1.000000e+02
METEMI	170.0	47876.133575	1.346611e+05	11.596147	3662.884908	11118.976025	32368.909040	1.186285e+06
FORARE	173.0	32.218177	2.312001e+01	0.008078	11.604388	31.509048	49.071780	9.741212e+01
CO2EMI	170.0	3.751097	4.606479e+00	0.032585	0.631924	2.298368	4.823496	3.172684e+01
PM2EXP	167.0	91.940595	2.206003e+01	0.274092	99.627134	100.000000	100.000000	1.000000e+02
POPDEN	174.0	200.886765	6.453834e+02	2.115134	27.454539	77.983133	153.993650	7.918951e+03
ENRTER	116.0	49.994997	2.970619e+01	2.432581	22.107195	53.392460	71.057467	1.433107e+02
GDPCAP	170.0	13992.095610	1.957954e+04	216.827417	1870.503029	5348.192875	17421.116227	1.173705e+05
EPISCO	165.0	42.946667	1.249086e+01	18.900000	33.000000	40.900000	50.500000	7.790000e+01

##################################
# Performing a general exploration of the object variable
##################################
print('Object Variable Summary:')
display(cancer_rate.describe(include='object').transpose())

Object Variable Summary:

	count	unique	top	freq
COUNTRY	177	177	Australia	1

##################################
# Performing a general exploration of the categorical variables
##################################
print('Categorical Variable Summary:')
display(cancer_rate.describe(include='category').transpose())

Categorical Variable Summary:

	count	unique	top	freq
CANRAT	177	2	Low	132
HDICAT	167	4	VH	59

##################################
# Performing a general exploration of the response variable
##################################
cancer_rate.CANRAT.value_counts(normalize = True)

Low     0.745763
High    0.254237
Name: CANRAT, dtype: float64

1.3. Data Quality Assessment

Data quality findings based on assessment are as follows:

1. No duplicated rows observed.
2. Missing data noted for 20 variables with Null.Count>0 and Fill.Rate<1.0.
   • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
   • PATRES: Null.Count = 69, Fill.Rate = 0.610
   • ENRTER: Null.Count = 61, Fill.Rate = 0.655
   • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • GDPPER: Null.Count = 12, Fill.Rate = 0.932
   • EPISCO: Null.Count = 12, Fill.Rate = 0.932
   • HDICAT: Null.Count = 10, Fill.Rate = 0.943
   • PM2EXP: Null.Count = 10, Fill.Rate = 0.943
   • DTHCMD: Null.Count = 7, Fill.Rate = 0.960
   • METEMI: Null.Count = 7, Fill.Rate = 0.960
   • CO2EMI: Null.Count = 7, Fill.Rate = 0.960
   • GDPCAP: Null.Count = 7, Fill.Rate = 0.960
   • GHGEMI: Null.Count = 7, Fill.Rate = 0.960
   • FORARE: Null.Count = 4, Fill.Rate = 0.977
   • TUBINC: Null.Count = 3, Fill.Rate = 0.983
   • AGRLND: Null.Count = 3, Fill.Rate = 0.983
   • POPGRO: Null.Count = 3, Fill.Rate = 0.983
   • POPDEN: Null.Count = 3, Fill.Rate = 0.983
   • URBPOP: Null.Count = 3, Fill.Rate = 0.983
   • LIFEXP: Null.Count = 3, Fill.Rate = 0.983
3. 120 observations noted with at least 1 missing data. From this number, 14 observations reported high Missing.Rate>0.2.
   • COUNTRY=Guadeloupe: Missing.Rate= 0.909
   • COUNTRY=Martinique: Missing.Rate= 0.909
   • COUNTRY=French Guiana: Missing.Rate= 0.909
   • COUNTRY=New Caledonia: Missing.Rate= 0.500
   • COUNTRY=French Polynesia: Missing.Rate= 0.500
   • COUNTRY=Guam: Missing.Rate= 0.500
   • COUNTRY=Puerto Rico: Missing.Rate= 0.409
   • COUNTRY=North Korea: Missing.Rate= 0.227
   • COUNTRY=Somalia: Missing.Rate= 0.227
   • COUNTRY=South Sudan: Missing.Rate= 0.227
   • COUNTRY=Venezuela: Missing.Rate= 0.227
   • COUNTRY=Libya: Missing.Rate= 0.227
   • COUNTRY=Eritrea: Missing.Rate= 0.227
   • COUNTRY=Yemen: Missing.Rate= 0.227
4. Low variance observed for 1 variable with First.Second.Mode.Ratio>5.
   • PM2EXP: First.Second.Mode.Ratio = 53.000
5. No low variance observed for any variable with Unique.Count.Ratio>10.
6. High skewness observed for 5 variables with Skewness>3 or Skewness<(-3).
   • POPDEN: Skewness = +10.267
   • GHGEMI: Skewness = +9.496
   • PATRES: Skewness = +9.284
   • METEMI: Skewness = +5.801
   • PM2EXP: Skewness = -3.141

##################################
# Counting the number of duplicated rows
##################################
cancer_rate.duplicated().sum()

0

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate)] * len(cancer_rate.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all columns
##################################
all_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                              data_type_list,
                                              row_count_list,
                                              non_null_count_list,
                                              null_count_list,
                                              fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(all_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	COUNTRY	object	177	177	0	1.000000
1	CANRAT	category	177	177	0	1.000000
2	GDPPER	float64	177	165	12	0.932203
3	URBPOP	float64	177	174	3	0.983051
4	PATRES	float64	177	108	69	0.610169
5	RNDGDP	float64	177	74	103	0.418079
6	POPGRO	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051
8	TUBINC	float64	177	174	3	0.983051
9	DTHCMD	float64	177	170	7	0.960452
10	AGRLND	float64	177	174	3	0.983051
11	GHGEMI	float64	177	170	7	0.960452
12	RELOUT	float64	177	153	24	0.864407
13	METEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
15	CO2EMI	float64	177	170	7	0.960452
16	PM2EXP	float64	177	167	10	0.943503
17	POPDEN	float64	177	174	3	0.983051
18	ENRTER	float64	177	116	61	0.655367
19	GDPCAP	float64	177	170	7	0.960452
20	HDICAT	category	177	167	10	0.943503
21	EPISCO	float64	177	165	12	0.932203

##################################
# Counting the number of columns
# with Fill.Rate < 1.00
##################################
len(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)])

20

##################################
# Identifying the columns
# with Fill.Rate < 1.00
##################################
display(all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<1)].sort_values(by=['Fill.Rate'], ascending=True))

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
5	RNDGDP	float64	177	74	103	0.418079
4	PATRES	float64	177	108	69	0.610169
18	ENRTER	float64	177	116	61	0.655367
12	RELOUT	float64	177	153	24	0.864407
2	GDPPER	float64	177	165	12	0.932203
21	EPISCO	float64	177	165	12	0.932203
20	HDICAT	category	177	167	10	0.943503
16	PM2EXP	float64	177	167	10	0.943503
9	DTHCMD	float64	177	170	7	0.960452
13	METEMI	float64	177	170	7	0.960452
15	CO2EMI	float64	177	170	7	0.960452
19	GDPCAP	float64	177	170	7	0.960452
11	GHGEMI	float64	177	170	7	0.960452
14	FORARE	float64	177	173	4	0.977401
8	TUBINC	float64	177	174	3	0.983051
10	AGRLND	float64	177	174	3	0.983051
6	POPGRO	float64	177	174	3	0.983051
17	POPDEN	float64	177	174	3	0.983051
3	URBPOP	float64	177	174	3	0.983051
7	LIFEXP	float64	177	174	3	0.983051

##################################
# Identifying the rows
# with Fill.Rate < 0.90
##################################
column_low_fill_rate = all_column_quality_summary[(all_column_quality_summary['Fill.Rate']<0.90)]

##################################
# Gathering the metadata labels for each observation
##################################
row_metadata_list = cancer_rate["COUNTRY"].values.tolist()

##################################
# Gathering the number of columns for each observation
##################################
column_count_list = list([len(cancer_rate.columns)] * len(cancer_rate))

##################################
# Gathering the number of missing data for each row
##################################
null_row_list = list(cancer_rate.isna().sum(axis=1))

##################################
# Gathering the missing data percentage for each column
##################################
missing_rate_list = map(truediv, null_row_list, column_count_list)

##################################
# Identifying the rows
# with missing data
##################################
all_row_quality_summary = pd.DataFrame(zip(row_metadata_list,
                                           column_count_list,
                                           null_row_list,
                                           missing_rate_list), 
                                        columns=['Row.Name',
                                                 'Column.Count',
                                                 'Null.Count',                                                 
                                                 'Missing.Rate'])
display(all_row_quality_summary)

	Row.Name	Column.Count	Null.Count	Missing.Rate
0	Australia	22	1	0.045455
1	New Zealand	22	2	0.090909
2	Ireland	22	0	0.000000
3	United States	22	0	0.000000
4	Denmark	22	0	0.000000
...	...	...	...	...
172	Congo Republic	22	3	0.136364
173	Bhutan	22	2	0.090909
174	Nepal	22	2	0.090909
175	Gambia	22	4	0.181818
176	Niger	22	2	0.090909

177 rows × 4 columns

##################################
# Counting the number of rows
# with Missing.Rate > 0.00
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.00)])

120

##################################
# Counting the number of rows
# with Missing.Rate > 0.20
##################################
len(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)])

14

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
row_high_missing_rate = all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)]

##################################
# Identifying the rows
# with Missing.Rate > 0.20
##################################
display(all_row_quality_summary[(all_row_quality_summary['Missing.Rate']>0.20)].sort_values(by=['Missing.Rate'], ascending=False))

	Row.Name	Column.Count	Null.Count	Missing.Rate
35	Guadeloupe	22	20	0.909091
39	Martinique	22	20	0.909091
56	French Guiana	22	20	0.909091
13	New Caledonia	22	11	0.500000
44	French Polynesia	22	11	0.500000
75	Guam	22	11	0.500000
53	Puerto Rico	22	9	0.409091
85	North Korea	22	6	0.272727
132	Somalia	22	6	0.272727
168	South Sudan	22	6	0.272727
73	Venezuela	22	5	0.227273
117	Libya	22	5	0.227273
161	Eritrea	22	5	0.227273
164	Yemen	22	5	0.227273

##################################
# Formulating the dataset
# with numeric columns only
##################################
cancer_rate_numeric = cancer_rate.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = cancer_rate_numeric.columns

##################################
# Gathering the minimum value for each numeric column
##################################
numeric_minimum_list = cancer_rate_numeric.min()

##################################
# Gathering the mean value for each numeric column
##################################
numeric_mean_list = cancer_rate_numeric.mean()

##################################
# Gathering the median value for each numeric column
##################################
numeric_median_list = cancer_rate_numeric.median()

##################################
# Gathering the maximum value for each numeric column
##################################
numeric_maximum_list = cancer_rate_numeric.max()

##################################
# Gathering the first mode values for each numeric column
##################################
numeric_first_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[0] for x in cancer_rate_numeric]

##################################
# Gathering the second mode values for each numeric column
##################################
numeric_second_mode_list = [cancer_rate[x].value_counts(dropna=True).index.tolist()[1] for x in cancer_rate_numeric]

##################################
# Gathering the count of first mode values for each numeric column
##################################
numeric_first_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the count of second mode values for each numeric column
##################################
numeric_second_mode_count_list = [cancer_rate_numeric[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_numeric]

##################################
# Gathering the first mode to second mode ratio for each numeric column
##################################
numeric_first_second_mode_ratio_list = map(truediv, numeric_first_mode_count_list, numeric_second_mode_count_list)

##################################
# Gathering the count of unique values for each numeric column
##################################
numeric_unique_count_list = cancer_rate_numeric.nunique(dropna=True)

##################################
# Gathering the number of observations for each numeric column
##################################
numeric_row_count_list = list([len(cancer_rate_numeric)] * len(cancer_rate_numeric.columns))

##################################
# Gathering the unique to count ratio for each numeric column
##################################
numeric_unique_count_ratio_list = map(truediv, numeric_unique_count_list, numeric_row_count_list)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_numeric.skew()

##################################
# Gathering the kurtosis value for each numeric column
##################################
numeric_kurtosis_list = cancer_rate_numeric.kurtosis()

numeric_column_quality_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                numeric_minimum_list,
                                                numeric_mean_list,
                                                numeric_median_list,
                                                numeric_maximum_list,
                                                numeric_first_mode_list,
                                                numeric_second_mode_list,
                                                numeric_first_mode_count_list,
                                                numeric_second_mode_count_list,
                                                numeric_first_second_mode_ratio_list,
                                                numeric_unique_count_list,
                                                numeric_row_count_list,
                                                numeric_unique_count_ratio_list,
                                                numeric_skewness_list,
                                                numeric_kurtosis_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Minimum',
                                                 'Mean',
                                                 'Median',
                                                 'Maximum',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio',
                                                 'Skewness',
                                                 'Kurtosis'])
display(numeric_column_quality_summary)

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
0	GDPPER	1718.804896	45284.424283	34024.900890	2.346469e+05	98380.636010	42154.178100	1	1	1.000000	165	177	0.932203	1.517574	3.471992
1	URBPOP	13.345000	59.788121	61.701500	1.000000e+02	100.000000	52.516000	2	1	2.000000	173	177	0.977401	-0.210702	-0.962847
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
3	RNDGDP	0.039770	1.197474	0.873660	5.354510e+00	1.232440	0.962180	1	1	1.000000	74	177	0.418079	1.396742	1.695957
4	POPGRO	-2.079337	1.127028	1.179959	3.727101e+00	1.235701	1.483129	1	1	1.000000	174	177	0.983051	-0.195161	-0.423580
5	LIFEXP	52.777000	71.746113	72.464610	8.456000e+01	83.200000	68.687000	1	1	1.000000	174	177	0.983051	-0.357965	-0.649601
6	TUBINC	0.770000	105.005862	44.500000	5.920000e+02	12.000000	7.200000	4	3	1.333333	131	177	0.740113	1.746333	2.429368
7	DTHCMD	1.283611	21.260521	12.456279	6.520789e+01	4.941054	42.079403	1	1	1.000000	170	177	0.960452	0.900509	-0.691541
8	AGRLND	0.512821	38.793456	40.386649	8.084112e+01	46.252480	72.006469	1	1	1.000000	174	177	0.983051	0.074000	-0.926249
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
10	RELOUT	0.000296	39.760036	32.381668	1.000000e+02	100.000000	13.637841	3	1	3.000000	151	177	0.853107	0.501088	-0.981774
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
12	FORARE	0.008078	32.218177	31.509048	9.741212e+01	17.421315	8.782159	1	1	1.000000	173	177	0.977401	0.519277	-0.322589
13	CO2EMI	0.032585	3.751097	2.298368	3.172684e+01	14.772658	0.972088	1	1	1.000000	170	177	0.960452	2.721552	10.311574
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
16	ENRTER	2.432581	49.994997	53.392460	1.433107e+02	110.139221	45.220661	1	1	1.000000	116	177	0.655367	0.275863	-0.392895
17	GDPCAP	216.827417	13992.095610	5348.192875	1.173705e+05	51722.069000	3961.726633	1	1	1.000000	170	177	0.960452	2.258568	5.938690
18	EPISCO	18.900000	42.946667	40.900000	7.790000e+01	29.600000	43.600000	3	3	1.000000	137	177	0.774011	0.641799	0.035208

##################################
# Counting the number of numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)])

1

##################################
# Identifying the numeric columns
# with First.Second.Mode.Ratio > 5.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['First.Second.Mode.Ratio']>5)].sort_values(by=['First.Second.Mode.Ratio'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
14	PM2EXP	0.274092	91.940595	100.0	100.0	100.0	100.0	106	2	53.0	61	177	0.344633	-3.141557	9.032386

##################################
# Counting the number of numeric columns
# with Unique.Count.Ratio > 10.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Counting the number of numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
len(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))])

5

##################################
# Identifying the numeric columns
# with Skewness > 3.00 or Skewness < -3.00
##################################
display(numeric_column_quality_summary[(numeric_column_quality_summary['Skewness']>3) | (numeric_column_quality_summary['Skewness']<(-3))].sort_values(by=['Skewness'], ascending=False))

	Numeric.Column.Name	Minimum	Mean	Median	Maximum	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio	Skewness	Kurtosis
15	POPDEN	2.115134	200.886765	77.983133	7.918951e+03	3.335312	13.300785	1	1	1.000000	174	177	0.983051	10.267750	119.995256
9	GHGEMI	179.725150	259582.709895	41009.275980	1.294287e+07	571903.119900	3000.932259	1	1	1.000000	170	177	0.960452	9.496120	101.637308
2	PATRES	1.000000	20607.388889	244.500000	1.344817e+06	6.000000	2.000000	4	3	1.333333	97	177	0.548023	9.284436	91.187178
11	METEMI	11.596147	47876.133575	11118.976025	1.186285e+06	131484.763200	1326.034028	1	1	1.000000	170	177	0.960452	5.801014	38.661386
14	PM2EXP	0.274092	91.940595	100.000000	1.000000e+02	100.000000	100.000000	106	2	53.000000	61	177	0.344633	-3.141557	9.032386

##################################
# Formulating the dataset
# with object column only
##################################
cancer_rate_object = cancer_rate.select_dtypes(include='object')

##################################
# Gathering the variable names for the object column
##################################
object_variable_name_list = cancer_rate_object.columns

##################################
# Gathering the first mode values for the object column
##################################
object_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_object]

##################################
# Gathering the second mode values for each object column
##################################
object_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_object]

##################################
# Gathering the count of first mode values for each object column
##################################
object_first_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_object]

##################################
# Gathering the count of second mode values for each object column
##################################
object_second_mode_count_list = [cancer_rate_object[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_object]

##################################
# Gathering the first mode to second mode ratio for each object column
##################################
object_first_second_mode_ratio_list = map(truediv, object_first_mode_count_list, object_second_mode_count_list)

##################################
# Gathering the count of unique values for each object column
##################################
object_unique_count_list = cancer_rate_object.nunique(dropna=True)

##################################
# Gathering the number of observations for each object column
##################################
object_row_count_list = list([len(cancer_rate_object)] * len(cancer_rate_object.columns))

##################################
# Gathering the unique to count ratio for each object column
##################################
object_unique_count_ratio_list = map(truediv, object_unique_count_list, object_row_count_list)

object_column_quality_summary = pd.DataFrame(zip(object_variable_name_list,
                                                 object_first_mode_list,
                                                 object_second_mode_list,
                                                 object_first_mode_count_list,
                                                 object_second_mode_count_list,
                                                 object_first_second_mode_ratio_list,
                                                 object_unique_count_list,
                                                 object_row_count_list,
                                                 object_unique_count_ratio_list), 
                                        columns=['Object.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(object_column_quality_summary)

	Object.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	COUNTRY	Australia	Mauritius	1	1	1.0	177	177	1.0

##################################
# Counting the number of object columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of object columns
# with Unique.Count.Ratio > 10.00
##################################
len(object_column_quality_summary[(object_column_quality_summary['Unique.Count.Ratio']>10)])

0

##################################
# Formulating the dataset
# with categorical columns only
##################################
cancer_rate_categorical = cancer_rate.select_dtypes(include='category')

##################################
# Gathering the variable names for the categorical column
##################################
categorical_variable_name_list = cancer_rate_categorical.columns

##################################
# Gathering the first mode values for each categorical column
##################################
categorical_first_mode_list = [cancer_rate[x].value_counts().index.tolist()[0] for x in cancer_rate_categorical]

##################################
# Gathering the second mode values for each categorical column
##################################
categorical_second_mode_list = [cancer_rate[x].value_counts().index.tolist()[1] for x in cancer_rate_categorical]

##################################
# Gathering the count of first mode values for each categorical column
##################################
categorical_first_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[0]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the count of second mode values for each categorical column
##################################
categorical_second_mode_count_list = [cancer_rate_categorical[x].isin([cancer_rate[x].value_counts(dropna=True).index.tolist()[1]]).sum() for x in cancer_rate_categorical]

##################################
# Gathering the first mode to second mode ratio for each categorical column
##################################
categorical_first_second_mode_ratio_list = map(truediv, categorical_first_mode_count_list, categorical_second_mode_count_list)

##################################
# Gathering the count of unique values for each categorical column
##################################
categorical_unique_count_list = cancer_rate_categorical.nunique(dropna=True)

##################################
# Gathering the number of observations for each categorical column
##################################
categorical_row_count_list = list([len(cancer_rate_categorical)] * len(cancer_rate_categorical.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
categorical_unique_count_ratio_list = map(truediv, categorical_unique_count_list, categorical_row_count_list)

categorical_column_quality_summary = pd.DataFrame(zip(categorical_variable_name_list,
                                                    categorical_first_mode_list,
                                                    categorical_second_mode_list,
                                                    categorical_first_mode_count_list,
                                                    categorical_second_mode_count_list,
                                                    categorical_first_second_mode_ratio_list,
                                                    categorical_unique_count_list,
                                                    categorical_row_count_list,
                                                    categorical_unique_count_ratio_list), 
                                        columns=['Categorical.Column.Name',
                                                 'First.Mode',
                                                 'Second.Mode',
                                                 'First.Mode.Count',
                                                 'Second.Mode.Count',
                                                 'First.Second.Mode.Ratio',
                                                 'Unique.Count',
                                                 'Row.Count',
                                                 'Unique.Count.Ratio'])
display(categorical_column_quality_summary)

	Categorical.Column.Name	First.Mode	Second.Mode	First.Mode.Count	Second.Mode.Count	First.Second.Mode.Ratio	Unique.Count	Row.Count	Unique.Count.Ratio
0	CANRAT	Low	High	132	45	2.933333	2	177	0.011299
1	HDICAT	VH	H	59	39	1.512821	4	177	0.022599

##################################
# Counting the number of categorical columns
# with First.Second.Mode.Ratio > 5.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['First.Second.Mode.Ratio']>5)])

0

##################################
# Counting the number of categorical columns
# with Unique.Count.Ratio > 10.00
##################################
len(categorical_column_quality_summary[(categorical_column_quality_summary['Unique.Count.Ratio']>10)])

0

1.4. Data Preprocessing

1.4.1 Data Cleaning

1. Subsets of rows and columns with high rates of missing data were removed from the dataset:
   • 4 variables with Fill.Rate<0.9 were excluded for subsequent analysis.
     • RNDGDP: Null.Count = 103, Fill.Rate = 0.418
     • PATRES: Null.Count = 69, Fill.Rate = 0.610
     • ENRTER: Null.Count = 61, Fill.Rate = 0.655
     • RELOUT: Null.Count = 24, Fill.Rate = 0.864
   • 14 rows with Missing.Rate>0.2 were exluded for subsequent analysis.
     • COUNTRY=Guadeloupe: Missing.Rate= 0.909
     • COUNTRY=Martinique: Missing.Rate= 0.909
     • COUNTRY=French Guiana: Missing.Rate= 0.909
     • COUNTRY=New Caledonia: Missing.Rate= 0.500
     • COUNTRY=French Polynesia: Missing.Rate= 0.500
     • COUNTRY=Guam: Missing.Rate= 0.500
     • COUNTRY=Puerto Rico: Missing.Rate= 0.409
     • COUNTRY=North Korea: Missing.Rate= 0.227
     • COUNTRY=Somalia: Missing.Rate= 0.227
     • COUNTRY=South Sudan: Missing.Rate= 0.227
     • COUNTRY=Venezuela: Missing.Rate= 0.227
     • COUNTRY=Libya: Missing.Rate= 0.227
     • COUNTRY=Eritrea: Missing.Rate= 0.227
     • COUNTRY=Yemen: Missing.Rate= 0.227
2. No variables were removed due to zero or near-zero variance.
3. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 15/18 predictor (numeric)
       • GDPPER
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • METEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/18 predictor (categorical)
       • HDICAT

##################################
# Performing a general exploration of the original dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate.shape)

Dataset Dimensions: 

(177, 22)

##################################
# Filtering out the rows with
# with Missing.Rate > 0.20
##################################
cancer_rate_filtered_row = cancer_rate.drop(cancer_rate[cancer_rate.COUNTRY.isin(row_high_missing_rate['Row.Name'].values.tolist())].index)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_filtered_row.shape)

Dataset Dimensions: 

(163, 22)

##################################
# Filtering out the columns with
# with Fill.Rate < 0.90
##################################
cancer_rate_filtered_row_column = cancer_rate_filtered_row.drop(column_low_fill_rate['Column.Name'].values.tolist(), axis=1)

##################################
# Formulating a new dataset object
# for the cleaned data
##################################
cancer_rate_cleaned = cancer_rate_filtered_row_column

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_cleaned.shape)

Dataset Dimensions: 

(163, 18)

1.4.2 Missing Data Imputation

Iterative Imputer is based on the Multivariate Imputation by Chained Equations (MICE) algorithm - an imputation method based on fully conditional specification, where each incomplete variable is imputed by a separate model. As a sequential regression imputation technique, the algorithm imputes an incomplete column (target column) by generating plausible synthetic values given other columns in the data. Each incomplete column must act as a target column, and has its own specific set of predictors. For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to prior to imputation of the target columns.

Linear Regression explores the linear relationship between a scalar response and one or more covariates by having the conditional mean of the dependent variable be an affine function of the independent variables. The relationship is modeled through a disturbance term which represents an unobserved random variable that adds noise. The algorithm is typically formulated from the data using the least squares method which seeks to estimate the coefficients by minimizing the squared residual function. The linear equation assigns one scale factor represented by a coefficient to each covariate and an additional coefficient called the intercept or the bias coefficient which gives the line an additional degree of freedom allowing to move up and down a two-dimensional plot.

1. Missing data for numeric variables were imputed using the iterative imputer algorithm with a linear regression estimator.
   • GDPPER: Null.Count = 1
   • FORARE: Null.Count = 1
   • PM2EXP: Null.Count = 5
2. Missing data for categorical variables were imputed using the most frequent value.
   • HDICAP: Null.Count = 1

##################################
# Formulating the summary
# for all cleaned columns
##################################
cleaned_column_quality_summary = pd.DataFrame(zip(list(cancer_rate_cleaned.columns),
                                                  list(cancer_rate_cleaned.dtypes),
                                                  list([len(cancer_rate_cleaned)] * len(cancer_rate_cleaned.columns)),
                                                  list(cancer_rate_cleaned.count()),
                                                  list(cancer_rate_cleaned.isna().sum(axis=0))), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count'])
display(cleaned_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count
0	COUNTRY	object	163	163	0
1	CANRAT	category	163	163	0
2	GDPPER	float64	163	162	1
3	URBPOP	float64	163	163	0
4	POPGRO	float64	163	163	0
5	LIFEXP	float64	163	163	0
6	TUBINC	float64	163	163	0
7	DTHCMD	float64	163	163	0
8	AGRLND	float64	163	163	0
9	GHGEMI	float64	163	163	0
10	METEMI	float64	163	163	0
11	FORARE	float64	163	162	1
12	CO2EMI	float64	163	163	0
13	PM2EXP	float64	163	158	5
14	POPDEN	float64	163	163	0
15	GDPCAP	float64	163	163	0
16	HDICAT	category	163	162	1
17	EPISCO	float64	163	163	0

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='object')

##################################
# Formulating the cleaned dataset
# with numeric columns only
##################################
cancer_rate_cleaned_numeric = cancer_rate_cleaned.select_dtypes(include='number')

##################################
# Taking a snapshot of the cleaned dataset
##################################
cancer_rate_cleaned_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	NaN	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Defining the estimator to be used
# at each step of the round-robin imputation
##################################
lr = LinearRegression()

##################################
# Defining the parameter of the
# iterative imputer which will estimate 
# the columns with missing values
# as a function of the other columns
# in a round-robin fashion
##################################
iterative_imputer = IterativeImputer(
    estimator = lr,
    max_iter = 10,
    tol = 1e-10,
    imputation_order = 'ascending',
    random_state=88888888
)

##################################
# Implementing the iterative imputer 
##################################
cancer_rate_imputed_numeric_array = iterative_imputer.fit_transform(cancer_rate_cleaned_numeric)

##################################
# Transforming the imputed data
# from an array to a dataframe
##################################
cancer_rate_imputed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array, 
                                           columns = cancer_rate_cleaned_numeric.columns)

##################################
# Taking a snapshot of the imputed dataset
##################################
cancer_rate_imputed_numeric.head()

	GDPPER	URBPOP	POPGRO	LIFEXP	TUBINC	DTHCMD	AGRLND	GHGEMI	METEMI	FORARE	CO2EMI	PM2EXP	POPDEN	GDPCAP	EPISCO
0	98380.63601	86.241	1.235701	83.200000	7.2	4.941054	46.252480	5.719031e+05	131484.763200	17.421315	14.772658	24.893584	3.335312	51722.06900	60.1
1	77541.76438	86.699	2.204789	82.256098	7.2	4.354730	38.562911	8.015803e+04	32241.937000	37.570126	6.160799	65.867296	19.331586	41760.59478	56.7
2	198405.87500	63.653	1.029111	82.556098	5.3	5.684596	65.495718	5.949773e+04	15252.824630	11.351720	6.768228	0.274092	72.367281	85420.19086	57.4
3	130941.63690	82.664	0.964348	76.980488	2.3	5.302060	44.363367	5.505181e+06	748241.402900	33.866926	13.032828	3.343170	36.240985	63528.63430	51.1
4	113300.60110	88.116	0.291641	81.602439	4.1	6.826140	65.499675	4.113555e+04	7778.773921	15.711000	4.691237	56.914456	145.785100	60915.42440	77.9

##################################
# Formulating the cleaned dataset
# with categorical columns only
##################################
cancer_rate_cleaned_categorical = cancer_rate_cleaned.select_dtypes(include='category')

##################################
# Imputing the missing data
# for categorical columns with
# the most frequent category
##################################
cancer_rate_cleaned_categorical['HDICAT'].fillna(cancer_rate_cleaned_categorical['HDICAT'].mode()[0], inplace=True)
cancer_rate_imputed_categorical = cancer_rate_cleaned_categorical.reset_index(drop=True)

##################################
# Formulating the imputed dataset
##################################
cancer_rate_imputed = pd.concat([cancer_rate_imputed_numeric,cancer_rate_imputed_categorical], axis=1, join='inner')  

##################################
# Gathering the data types for each column
##################################
data_type_list = list(cancer_rate_imputed.dtypes)

##################################
# Gathering the variable names for each column
##################################
variable_name_list = list(cancer_rate_imputed.columns)

##################################
# Gathering the number of observations for each column
##################################
row_count_list = list([len(cancer_rate_imputed)] * len(cancer_rate_imputed.columns))

##################################
# Gathering the number of missing data for each column
##################################
null_count_list = list(cancer_rate_imputed.isna().sum(axis=0))

##################################
# Gathering the number of non-missing data for each column
##################################
non_null_count_list = list(cancer_rate_imputed.count())

##################################
# Gathering the missing data percentage for each column
##################################
fill_rate_list = map(truediv, non_null_count_list, row_count_list)

##################################
# Formulating the summary
# for all imputed columns
##################################
imputed_column_quality_summary = pd.DataFrame(zip(variable_name_list,
                                                  data_type_list,
                                                  row_count_list,
                                                  non_null_count_list,
                                                  null_count_list,
                                                  fill_rate_list), 
                                        columns=['Column.Name',
                                                 'Column.Type',
                                                 'Row.Count',
                                                 'Non.Null.Count',
                                                 'Null.Count',                                                 
                                                 'Fill.Rate'])
display(imputed_column_quality_summary)

	Column.Name	Column.Type	Row.Count	Non.Null.Count	Null.Count	Fill.Rate
0	GDPPER	float64	163	163	0	1.0
1	URBPOP	float64	163	163	0	1.0
2	POPGRO	float64	163	163	0	1.0
3	LIFEXP	float64	163	163	0	1.0
4	TUBINC	float64	163	163	0	1.0
5	DTHCMD	float64	163	163	0	1.0
6	AGRLND	float64	163	163	0	1.0
7	GHGEMI	float64	163	163	0	1.0
8	METEMI	float64	163	163	0	1.0
9	FORARE	float64	163	163	0	1.0
10	CO2EMI	float64	163	163	0	1.0
11	PM2EXP	float64	163	163	0	1.0
12	POPDEN	float64	163	163	0	1.0
13	GDPCAP	float64	163	163	0	1.0
14	EPISCO	float64	163	163	0	1.0
15	CANRAT	category	163	163	0	1.0
16	HDICAT	category	163	163	0	1.0

1.4.3 Outlier Detection

1. High number of outliers observed for 5 numeric variables with Outlier.Ratio>0.10 and marginal to high Skewness.
   • PM2EXP: Outlier.Count = 37, Outlier.Ratio = 0.226, Skewness=-3.061
   • GHGEMI: Outlier.Count = 27, Outlier.Ratio = 0.165, Skewness=+9.299
   • GDPCAP: Outlier.Count = 22, Outlier.Ratio = 0.134, Skewness=+2.311
   • POPDEN: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+9.972
   • METEMI: Outlier.Count = 20, Outlier.Ratio = 0.122, Skewness=+5.688
2. Minimal number of outliers observed for 5 numeric variables with Outlier.Ratio<0.10 and normal Skewness.
   • TUBINC: Outlier.Count = 12, Outlier.Ratio = 0.073, Skewness=+1.747
   • CO2EMI: Outlier.Count = 11, Outlier.Ratio = 0.067, Skewness=+2.693
   • GDPPER: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+1.554
   • EPISCO: Outlier.Count = 3, Outlier.Ratio = 0.018, Skewness=+0.635
   • CANRAT: Outlier.Count = 2, Outlier.Ratio = 0.012, Skewness=+0.910

##################################
# Formulating the imputed dataset
# with numeric columns only
##################################
cancer_rate_imputed_numeric = cancer_rate_imputed.select_dtypes(include='number')

##################################
# Gathering the variable names for each numeric column
##################################
numeric_variable_name_list = list(cancer_rate_imputed_numeric.columns)

##################################
# Gathering the skewness value for each numeric column
##################################
numeric_skewness_list = cancer_rate_imputed_numeric.skew()

##################################
# Computing the interquartile range
# for all columns
##################################
cancer_rate_imputed_numeric_q1 = cancer_rate_imputed_numeric.quantile(0.25)
cancer_rate_imputed_numeric_q3 = cancer_rate_imputed_numeric.quantile(0.75)
cancer_rate_imputed_numeric_iqr = cancer_rate_imputed_numeric_q3 - cancer_rate_imputed_numeric_q1

##################################
# Gathering the outlier count for each numeric column
# based on the interquartile range criterion
##################################
numeric_outlier_count_list = ((cancer_rate_imputed_numeric < (cancer_rate_imputed_numeric_q1 - 1.5 * cancer_rate_imputed_numeric_iqr)) | (cancer_rate_imputed_numeric > (cancer_rate_imputed_numeric_q3 + 1.5 * cancer_rate_imputed_numeric_iqr))).sum()

##################################
# Gathering the number of observations for each column
##################################
numeric_row_count_list = list([len(cancer_rate_imputed_numeric)] * len(cancer_rate_imputed_numeric.columns))

##################################
# Gathering the unique to count ratio for each categorical column
##################################
numeric_outlier_ratio_list = map(truediv, numeric_outlier_count_list, numeric_row_count_list)

##################################
# Formulating the outlier summary
# for all numeric columns
##################################
numeric_column_outlier_summary = pd.DataFrame(zip(numeric_variable_name_list,
                                                  numeric_skewness_list,
                                                  numeric_outlier_count_list,
                                                  numeric_row_count_list,
                                                  numeric_outlier_ratio_list), 
                                        columns=['Numeric.Column.Name',
                                                 'Skewness',
                                                 'Outlier.Count',
                                                 'Row.Count',
                                                 'Outlier.Ratio'])
display(numeric_column_outlier_summary)

	Numeric.Column.Name	Skewness	Outlier.Count	Row.Count	Outlier.Ratio
0	GDPPER	1.554457	3	163	0.018405
1	URBPOP	-0.212327	0	163	0.000000
2	POPGRO	-0.181666	0	163	0.000000
3	LIFEXP	-0.329704	0	163	0.000000
4	TUBINC	1.747962	12	163	0.073620
5	DTHCMD	0.930709	0	163	0.000000
6	AGRLND	0.035315	0	163	0.000000
7	GHGEMI	9.299960	27	163	0.165644
8	METEMI	5.688689	20	163	0.122699
9	FORARE	0.563015	0	163	0.000000
10	CO2EMI	2.693585	11	163	0.067485
11	PM2EXP	-3.088403	37	163	0.226994
12	POPDEN	9.972806	20	163	0.122699
13	GDPCAP	2.311079	22	163	0.134969
14	EPISCO	0.635994	3	163	0.018405

##################################
# Formulating the individual boxplots
# for all numeric columns
##################################
for column in cancer_rate_imputed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_imputed_numeric, x=column)

1.4.4 Collinearity

Pearson’s Correlation Coefficient is a parametric measure of the linear correlation for a pair of features by calculating the ratio between their covariance and the product of their standard deviations. The presence of high absolute correlation values indicate the univariate association between the numeric predictors and the numeric response.

1. Majority of the numeric variables reported moderate to high correlation which were statistically significant.
2. Among pairwise combinations of numeric variables, high Pearson.Correlation.Coefficient values were noted for:
   • GDPPER and GDPCAP: Pearson.Correlation.Coefficient = +0.921
   • GHGEMI and METEMI: Pearson.Correlation.Coefficient = +0.905
3. Among the highly correlated pairs, variables with the lowest correlation against the target variable were removed.
   • GDPPER: Pearson.Correlation.Coefficient = +0.690
   • METEMI: Pearson.Correlation.Coefficient = +0.062
4. The cleaned dataset is comprised of:
   • 163 rows (observations)
   • 16 columns (variables)
     • 1/16 metadata (object)
       • COUNTRY
     • 1/16 target (categorical)
       • CANRAT
     • 13/16 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • PM2EXP
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/16 predictor (categorical)
       • HDICAT

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
def plot_correlation_matrix(corr, mask=None):
    f, ax = plt.subplots(figsize=(11, 9))
    sns.heatmap(corr, 
                ax=ax,
                mask=mask,
                annot=True, 
                vmin=-1, 
                vmax=1, 
                center=0,
                cmap='coolwarm', 
                linewidths=1, 
                linecolor='gray', 
                cbar_kws={'orientation': 'horizontal'})  

##################################
# Computing the correlation coefficients
# and correlation p-values
# among pairs of numeric columns
##################################
cancer_rate_imputed_numeric_correlation_pairs = {}
cancer_rate_imputed_numeric_columns = cancer_rate_imputed_numeric.columns.tolist()
for numeric_column_a, numeric_column_b in itertools.combinations(cancer_rate_imputed_numeric_columns, 2):
    cancer_rate_imputed_numeric_correlation_pairs[numeric_column_a + '_' + numeric_column_b] = stats.pearsonr(
        cancer_rate_imputed_numeric.loc[:, numeric_column_a], 
        cancer_rate_imputed_numeric.loc[:, numeric_column_b])

##################################
# Formulating the pairwise correlation summary
# for all numeric columns
##################################
cancer_rate_imputed_numeric_summary = cancer_rate_imputed_numeric.from_dict(cancer_rate_imputed_numeric_correlation_pairs, orient='index')
cancer_rate_imputed_numeric_summary.columns = ['Pearson.Correlation.Coefficient', 'Correlation.PValue']
display(cancer_rate_imputed_numeric_summary.sort_values(by=['Pearson.Correlation.Coefficient'], ascending=False).head(20))

	Pearson.Correlation.Coefficient	Correlation.PValue
GDPPER_GDPCAP	0.921010	8.158179e-68
GHGEMI_METEMI	0.905121	1.087643e-61
POPGRO_DTHCMD	0.759470	7.124695e-32
GDPPER_LIFEXP	0.755787	2.055178e-31
GDPCAP_EPISCO	0.696707	5.312642e-25
LIFEXP_GDPCAP	0.683834	8.321371e-24
GDPPER_EPISCO	0.680812	1.555304e-23
GDPPER_URBPOP	0.666394	2.781623e-22
GDPPER_CO2EMI	0.654958	2.450029e-21
TUBINC_DTHCMD	0.643615	1.936081e-20
URBPOP_LIFEXP	0.623997	5.669778e-19
LIFEXP_EPISCO	0.620271	1.048393e-18
URBPOP_GDPCAP	0.559181	8.624533e-15
CO2EMI_GDPCAP	0.550221	2.782997e-14
URBPOP_CO2EMI	0.550046	2.846393e-14
LIFEXP_CO2EMI	0.531305	2.951829e-13
URBPOP_EPISCO	0.510131	3.507463e-12
POPGRO_TUBINC	0.442339	3.384403e-09
DTHCMD_PM2EXP	0.283199	2.491837e-04
CO2EMI_EPISCO	0.282734	2.553620e-04

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
##################################
cancer_rate_imputed_numeric_correlation = cancer_rate_imputed_numeric.corr()
mask = np.triu(cancer_rate_imputed_numeric_correlation)
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)
plt.show()

##################################
# Formulating a function 
# to plot the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
def correlation_significance(df=None):
    p_matrix = np.zeros(shape=(df.shape[1],df.shape[1]))
    for col in df.columns:
        for col2 in df.drop(col,axis=1).columns:
            _ , p = stats.pearsonr(df[col],df[col2])
            p_matrix[df.columns.to_list().index(col),df.columns.to_list().index(col2)] = p
    return p_matrix

##################################
# Plotting the correlation matrix
# for all pairwise combinations
# of numeric columns
# with significant p-values only
##################################
cancer_rate_imputed_numeric_correlation_p_values = correlation_significance(cancer_rate_imputed_numeric)                     
mask = np.invert(np.tril(cancer_rate_imputed_numeric_correlation_p_values<0.05)) 
plot_correlation_matrix(cancer_rate_imputed_numeric_correlation,mask)  

##################################
# Filtering out one among the 
# highly correlated variable pairs with
# lesser Pearson.Correlation.Coefficient
# when compared to the target variable
##################################
cancer_rate_imputed_numeric.drop(['GDPPER','METEMI'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_imputed_numeric.shape)

Dataset Dimensions: 

(163, 13)

1.4.5 Shape Transformation

Yeo-Johnson Transformation applies a new family of distributions that can be used without restrictions, extending many of the good properties of the Box-Cox power family. Similar to the Box-Cox transformation, the method also estimates the optimal value of lambda but has the ability to transform both positive and negative values by inflating low variance data and deflating high variance data to create a more uniform data set. While there are no restrictions in terms of the applicable values, the interpretability of the transformed values is more diminished as compared to the other methods.

1. A Yeo-Johnson transformation was applied to all numeric variables to improve distributional shape.
2. Most variables achieved symmetrical distributions with minimal outliers after transformation.
3. One variable which remained skewed even after applying shape transformation was removed.
   • PM2EXP
4. The transformed dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting a Yeo-Johnson Transformation
# to address the distributional
# shape of the variables
##################################
yeo_johnson_transformer = PowerTransformer(method='yeo-johnson',
                                          standardize=False)
cancer_rate_imputed_numeric_array = yeo_johnson_transformer.fit_transform(cancer_rate_imputed_numeric)

##################################
# Formulating a new dataset object
# for the transformed data
##################################
cancer_rate_transformed_numeric = pd.DataFrame(cancer_rate_imputed_numeric_array,
                                               columns=cancer_rate_imputed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_transformed_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_transformed_numeric, x=column)

##################################
# Filtering out the column
# which remained skewed even
# after applying shape transformation
##################################
cancer_rate_transformed_numeric.drop(['PM2EXP'], inplace=True, axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_transformed_numeric.shape)

Dataset Dimensions: 

(163, 12)

1.4.6 Centering and Scaling

1. All numeric variables were transformed using the standardization method to achieve a comparable scale between values.
2. The scaled dataset is comprised of:
   • 163 rows (observations)
   • 15 columns (variables)
     • 1/15 metadata (object)
       • COUNTRY
     • 1/15 target (categorical)
       • CANRAT
     • 12/15 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 1/15 predictor (categorical)
       • HDICAT

##################################
# Conducting standardization
# to transform the values of the 
# variables into comparable scale
##################################
standardization_scaler = StandardScaler()
cancer_rate_transformed_numeric_array = standardization_scaler.fit_transform(cancer_rate_transformed_numeric)

##################################
# Formulating a new dataset object
# for the scaled data
##################################
cancer_rate_scaled_numeric = pd.DataFrame(cancer_rate_transformed_numeric_array,
                                          columns=cancer_rate_transformed_numeric.columns)

##################################
# Formulating the individual boxplots
# for all transformed numeric columns
##################################
for column in cancer_rate_scaled_numeric:
        plt.figure(figsize=(17,1))
        sns.boxplot(data=cancer_rate_scaled_numeric, x=column)

1.4.7 Data Encoding

1. One-hot encoding was applied to the HDICAP_VH variable resulting to 4 additional columns in the dataset:
   • HDICAP_L
   • HDICAP_M
   • HDICAP_H
   • HDICAP_VH

##################################
# Formulating the categorical column
# for encoding transformation
##################################
cancer_rate_categorical_encoded = pd.DataFrame(cancer_rate_cleaned_categorical.loc[:, 'HDICAT'].to_list(),
                                               columns=['HDICAT'])

##################################
# Applying a one-hot encoding transformation
# for the categorical column
##################################
cancer_rate_categorical_encoded = pd.get_dummies(cancer_rate_categorical_encoded, columns=['HDICAT'])

1.4.8 Preprocessed Data Description

1. The preprocessed dataset is comprised of:
   • 163 rows (observations)
   • 18 columns (variables)
     • 1/18 metadata (object)
       • COUNTRY
     • 1/18 target (categorical)
       • CANRAT
     • 12/18 predictor (numeric)
       • URBPOP
       • POPGRO
       • LIFEXP
       • TUBINC
       • DTHCMD
       • AGRLND
       • GHGEMI
       • FORARE
       • CO2EMI
       • POPDEN
       • GDPCAP
       • EPISCO
     • 4/18 predictor (categorical)
       • HDICAT_L
       • HDICAT_M
       • HDICAT_H
       • HDICAT_VH

##################################
# Consolidating both numeric columns
# and encoded categorical columns
##################################
cancer_rate_preprocessed = pd.concat([cancer_rate_scaled_numeric,cancer_rate_categorical_encoded], axis=1, join='inner')  

##################################
# Performing a general exploration of the consolidated dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_preprocessed.shape)

Dataset Dimensions: 

(163, 16)

1.5. Data Exploration

1.5.1 Exploratory Data Analysis

1. Bivariate analysis identified individual predictors with generally positive association to the target variable based on visual inspection.
2. Higher values or higher proportions for the following predictors are associated with the CANRAT HIGH category:
   • URBPOP
   • LIFEXP
   • CO2EMI
   • GDPCAP
   • EPISCO
   • HDICAP_VH=1
3. Decreasing values or smaller proportions for the following predictors are associated with the CANRAT LOW category:
   • POPGRO
   • TUBINC
   • DTHCMD
   • HDICAP_L=0
   • HDICAP_M=0
   • HDICAP_H=0
4. Values for the following predictors are not associated with the CANRAT HIGH or LOW categories:
   • AGRLND
   • GHGEMI
   • FORARE
   • POPDEN

##################################
# Segregating the target
# and predictor variable lists
##################################
cancer_rate_preprocessed_target = cancer_rate_filtered_row['CANRAT'].to_frame()
cancer_rate_preprocessed_target.reset_index(inplace=True, drop=True)
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed[cancer_rate_categorical_encoded.columns]
cancer_rate_preprocessed_categorical_combined = cancer_rate_preprocessed_categorical.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed = cancer_rate_preprocessed.drop(cancer_rate_categorical_encoded.columns, axis=1) 
cancer_rate_preprocessed_predictors = cancer_rate_preprocessed.columns
cancer_rate_preprocessed_combined = cancer_rate_preprocessed.join(cancer_rate_preprocessed_target)
cancer_rate_preprocessed_all = cancer_rate_preprocessed_combined.join(cancer_rate_categorical_encoded)

##################################
# Segregating the target
# and predictor variable names
##################################
y_variable = 'CANRAT'
x_variables = cancer_rate_preprocessed_predictors

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 6
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 30))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual boxplots
# for all scaled numeric columns
##################################
for i, x_variable in enumerate(x_variables):
    ax = axes[i]
    ax.boxplot([group[x_variable] for name, group in cancer_rate_preprocessed_combined.groupby(y_variable)])
    ax.set_title(f'{y_variable} Versus {x_variable}')
    ax.set_xlabel(y_variable)
    ax.set_ylabel(x_variable)
    ax.set_xticks(range(1, len(cancer_rate_preprocessed_combined[y_variable].unique()) + 1), ['Low', 'High'])

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

##################################
# Segregating the target
# and predictor variable names
##################################
y_variables = cancer_rate_preprocessed_categorical.columns
x_variable = 'CANRAT'

##################################
# Defining the number of 
# rows and columns for the subplots
##################################
num_rows = 2
num_cols = 2

##################################
# Formulating the subplot structure
##################################
fig, axes = plt.subplots(num_rows, num_cols, figsize=(15, 10))

##################################
# Flattening the multi-row and
# multi-column axes
##################################
axes = axes.ravel()

##################################
# Formulating the individual stacked column plots
# for all categorical columns
##################################
for i, y_variable in enumerate(y_variables):
    ax = axes[i]
    category_counts = cancer_rate_preprocessed_categorical_combined.groupby([x_variable, y_variable]).size().unstack(fill_value=0)
    category_proportions = category_counts.div(category_counts.sum(axis=1), axis=0)
    category_proportions.plot(kind='bar', stacked=True, ax=ax)
    ax.set_title(f'{x_variable} Versus {y_variable}')
    ax.set_xlabel(x_variable)
    ax.set_ylabel('Proportions')

##################################
# Adjusting the subplot layout
##################################
plt.tight_layout()

##################################
# Presenting the subplots
##################################
plt.show()

1.5.2 Hypothesis Testing

1. The relationship between the numeric predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: Difference in the means between groups LOW and HIGH is equal to zero
   • Alternative: Difference in the means between groups LOW and HIGH is not equal to zero
2. There is sufficient evidence to conclude of a statistically significant difference between the means of the numeric measurements obtained from LOW and HIGH groups of the CANRAT target variable in 9 of the 12 numeric predictors given their high t-test statistic values with reported low p-values less than the significance level of 0.05.
   • GDPCAP: T.Test.Statistic=-11.937, Correlation.PValue=0.000
   • EPISCO: T.Test.Statistic=-11.789, Correlation.PValue=0.000
   • LIFEXP: T.Test.Statistic=-10.979, Correlation.PValue=0.000
   • TUBINC: T.Test.Statistic=+9.609, Correlation.PValue=0.000
   • DTHCMD: T.Test.Statistic=+8.376, Correlation.PValue=0.000
   • CO2EMI: T.Test.Statistic=-7.031, Correlation.PValue=0.000
   • URBPOP: T.Test.Statistic=-6.541, Correlation.PValue=0.000
   • POPGRO: T.Test.Statistic=+4.905, Correlation.PValue=0.000
   • GHGEMI: T.Test.Statistic=-2.243, Correlation.PValue=0.026
3. The relationship between the categorical predictors to the CANRAT target variable was statistically evaluated using the following hypotheses:
   • Null: The categorical predictor is independent of the categorical target variable
   • Alternative: The categorical predictor is dependent of the categorical target variable
4. There is sufficient evidence to conclude of a statistically significant relationship difference between the categories of the categorical predictors and the LOW and HIGH groups of the CANRAT target variable in all 4 categorical predictors given their high chisquare statistic values with reported low p-values less than the significance level of 0.05.
   • HDICAT_VH: ChiSquare.Test.Statistic=76.764, ChiSquare.Test.PValue=0.000
   • HDICAT_H: ChiSquare.Test.Statistic=13.860, ChiSquare.Test.PValue=0.000
   • HDICAT_M: ChiSquare.Test.Statistic=10.286, ChiSquare.Test.PValue=0.001
   • HDICAT_L: ChiSquare.Test.Statistic=9.081, ChiSquare.Test.PValue=0.002

##################################
# Computing the t-test 
# statistic and p-values
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_ttest_target = {}
cancer_rate_preprocessed_numeric = cancer_rate_preprocessed_combined
cancer_rate_preprocessed_numeric_columns = cancer_rate_preprocessed_predictors
for numeric_column in cancer_rate_preprocessed_numeric_columns:
    group_0 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='Low']
    group_1 = cancer_rate_preprocessed_numeric[cancer_rate_preprocessed_numeric.loc[:,'CANRAT']=='High']
    cancer_rate_preprocessed_numeric_ttest_target['CANRAT_' + numeric_column] = stats.ttest_ind(
        group_0[numeric_column], 
        group_1[numeric_column], 
        equal_var=True)

##################################
# Formulating the pairwise ttest summary
# between the target variable
# and numeric predictor columns
##################################
cancer_rate_preprocessed_numeric_summary = cancer_rate_preprocessed_numeric.from_dict(cancer_rate_preprocessed_numeric_ttest_target, orient='index')
cancer_rate_preprocessed_numeric_summary.columns = ['T.Test.Statistic', 'T.Test.PValue']
display(cancer_rate_preprocessed_numeric_summary.sort_values(by=['T.Test.PValue'], ascending=True).head(12))

	T.Test.Statistic	T.Test.PValue
CANRAT_GDPCAP	-11.936988	6.247937e-24
CANRAT_EPISCO	-11.788870	1.605980e-23
CANRAT_LIFEXP	-10.979098	2.754214e-21
CANRAT_TUBINC	9.608760	1.463678e-17
CANRAT_DTHCMD	8.375558	2.552108e-14
CANRAT_CO2EMI	-7.030702	5.537463e-11
CANRAT_URBPOP	-6.541001	7.734940e-10
CANRAT_POPGRO	4.904817	2.269446e-06
CANRAT_GHGEMI	-2.243089	2.625563e-02
CANRAT_FORARE	-1.174143	2.420717e-01
CANRAT_POPDEN	-0.495221	6.211191e-01
CANRAT_AGRLND	-0.047628	9.620720e-01

##################################
# Computing the chisquare
# statistic and p-values
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_chisquare_target = {}
cancer_rate_preprocessed_categorical = cancer_rate_preprocessed_categorical_combined
cancer_rate_preprocessed_categorical_columns = ['HDICAT_L','HDICAT_M','HDICAT_H','HDICAT_VH']
for categorical_column in cancer_rate_preprocessed_categorical_columns:
    contingency_table = pd.crosstab(cancer_rate_preprocessed_categorical[categorical_column], 
                                    cancer_rate_preprocessed_categorical['CANRAT'])
    cancer_rate_preprocessed_categorical_chisquare_target['CANRAT_' + categorical_column] = stats.chi2_contingency(
        contingency_table)[0:2]

##################################
# Formulating the pairwise chisquare summary
# between the target variable
# and categorical predictor columns
##################################
cancer_rate_preprocessed_categorical_summary = cancer_rate_preprocessed_categorical.from_dict(cancer_rate_preprocessed_categorical_chisquare_target, orient='index')
cancer_rate_preprocessed_categorical_summary.columns = ['ChiSquare.Test.Statistic', 'ChiSquare.Test.PValue']
display(cancer_rate_preprocessed_categorical_summary.sort_values(by=['ChiSquare.Test.PValue'], ascending=True).head(4))

	ChiSquare.Test.Statistic	ChiSquare.Test.PValue
CANRAT_HDICAT_VH	76.764134	1.926446e-18
CANRAT_HDICAT_M	13.860367	1.969074e-04
CANRAT_HDICAT_L	10.285575	1.340742e-03
CANRAT_HDICAT_H	9.080788	2.583087e-03

1.6. Model Development With Hyperparameter Tuning

Hyperparameter tuning is an iterative process that involves experimenting with different hyperparameter combinations, evaluating the model's performance, and refining the hyperparameter values to achieve the best possible performance on new, unseen data - aimed at building effective and well-generalizing machine learning models. A model's performance depends not only on the learned parameters (weights) during training but also on hyperparameters, which are external configuration settings that cannot be learned from the data.

1.6.1 Premodelling Data Description

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling, kind='reg')
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.6.2 Logistic Regression

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
optimal_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Logistic Regression model
##################################
optimal_logistic_regression.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Logistic Regression model
##################################
optimal_logistic_regression.best_score_ 
optimal_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': None,
 'penalty': 'l1',
 'random_state': 88888888,
 'solver': 'liblinear'}

##################################
# Evaluating the optimal logistic regression model
# on the train set
##################################
optimal_logistic_regression_y_hat_train = optimal_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_train = model_performance_evaluation(y_train, optimal_logistic_regression_y_hat_train)
optimal_logistic_regression_performance_train['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_train['set'] = ['train'] * 5
print('Optimal Logistic Regression Model Performance on Train Data: ')
display(optimal_logistic_regression_performance_train)

Optimal Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_logistic_regression	train
1	Precision	0.870968	optimal_logistic_regression	train
2	Recall	0.931034	optimal_logistic_regression	train
3	F1	0.900000	optimal_logistic_regression	train
4	AUROC	0.941988	optimal_logistic_regression	train

##################################
# Evaluating the optimal logistic regression model
# on the test set
##################################
optimal_logistic_regression_y_hat_test = optimal_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_logistic_regression_performance_test = model_performance_evaluation(y_test, optimal_logistic_regression_y_hat_test)
optimal_logistic_regression_performance_test['model'] = ['optimal_logistic_regression'] * 5
optimal_logistic_regression_performance_test['set'] = ['test'] * 5
print('Optimal Logistic Regression Model Performance on Test Data: ')
display(optimal_logistic_regression_performance_test)

Optimal Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	optimal_logistic_regression	test
1	Precision	0.888889	optimal_logistic_regression	test
2	Recall	0.666667	optimal_logistic_regression	test
3	F1	0.761905	optimal_logistic_regression	test
4	AUROC	0.819820	optimal_logistic_regression	test

1.6.3 Decision Trees

Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
optimal_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Decision Tree model
##################################
optimal_decision_tree.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Decision Tree model
##################################
optimal_decision_tree.best_score_ 
optimal_decision_tree.best_params_

{'class_weight': None,
 'criterion': 'entropy',
 'max_depth': 5,
 'min_samples_leaf': 3,
 'random_state': 88888888}

##################################
# Evaluating the optimal decision tree model
# on the train set
##################################
optimal_decision_tree_y_hat_train = optimal_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_train = model_performance_evaluation(y_train, optimal_decision_tree_y_hat_train)
optimal_decision_tree_performance_train['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_train['set'] = ['train'] * 5
print('Optimal Decision Tree Model Performance on Train Data: ')
display(optimal_decision_tree_performance_train)

Optimal Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.973684	optimal_decision_tree	train
1	Precision	1.000000	optimal_decision_tree	train
2	Recall	0.896552	optimal_decision_tree	train
3	F1	0.945455	optimal_decision_tree	train
4	AUROC	0.948276	optimal_decision_tree	train

##################################
# Evaluating the optimal decision tree model
# on the test set
##################################
optimal_decision_tree_y_hat_test = optimal_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_decision_tree_performance_test = model_performance_evaluation(y_test, optimal_decision_tree_y_hat_test)
optimal_decision_tree_performance_test['model'] = ['optimal_decision_tree'] * 5
optimal_decision_tree_performance_test['set'] = ['test'] * 5
print('Optimal Decision Tree Model Performance on Test Data: ')
display(optimal_decision_tree_performance_test)

Optimal Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.857143	optimal_decision_tree	test
1	Precision	0.857143	optimal_decision_tree	test
2	Recall	0.500000	optimal_decision_tree	test
3	F1	0.631579	optimal_decision_tree	test
4	AUROC	0.736486	optimal_decision_tree	test

1.6.4 Random Forest

Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [3,5,7],
    'max_features':['sqrt', 'log2'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
optimal_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Random Forest model
##################################
optimal_random_forest.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Random Forest model
##################################
optimal_random_forest.best_score_ 
optimal_random_forest.best_params_

{'class_weight': None,
 'criterion': 'gini',
 'max_depth': 5,
 'max_features': 'log2',
 'min_samples_leaf': 3,
 'n_estimators': 7,
 'random_state': 88888888}

##################################
# Evaluating the optimal Random Forest model
# on the train set
##################################
optimal_random_forest_y_hat_train = optimal_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_train = model_performance_evaluation(y_train, optimal_random_forest_y_hat_train)
optimal_random_forest_performance_train['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_train['set'] = ['train'] * 5
print('Optimal Random Forest Model Performance on Train Data: ')
display(optimal_random_forest_performance_train)

Optimal Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.964912	optimal_random_forest	train
1	Precision	0.931034	optimal_random_forest	train
2	Recall	0.931034	optimal_random_forest	train
3	F1	0.931034	optimal_random_forest	train
4	AUROC	0.953753	optimal_random_forest	train

##################################
# Evaluating the optimal Random Forest model
# on the test set
##################################
optimal_random_forest_y_hat_test = optimal_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_random_forest_performance_test = model_performance_evaluation(y_test, optimal_random_forest_y_hat_test)
optimal_random_forest_performance_test['model'] = ['optimal_random_forest'] * 5
optimal_random_forest_performance_test['set'] = ['test'] * 5
print('Optimal Random Forest Model Performance on Test Data: ')
display(optimal_random_forest_performance_test)

Optimal Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	optimal_random_forest	test
1	Precision	0.888889	optimal_random_forest	test
2	Recall	0.666667	optimal_random_forest	test
3	F1	0.761905	optimal_random_forest	test
4	AUROC	0.819820	optimal_random_forest	test

1.6.5 Support Vector Machine

Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [None],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
optimal_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the optimal Support Vector Machine model
##################################
optimal_support_vector_machine.fit(X_train, y_train)

##################################
# Determining the optimal hyperparameter
# for the Support Vector Machine model
##################################
optimal_support_vector_machine.best_score_ 
optimal_support_vector_machine.best_params_

{'C': 1.0, 'class_weight': None, 'kernel': 'poly', 'random_state': 88888888}

##################################
# Evaluating the optimal Support Vector Machine model
# on the train set
##################################
optimal_support_vector_machine_y_hat_train = optimal_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_train = model_performance_evaluation(y_train, optimal_support_vector_machine_y_hat_train)
optimal_support_vector_machine_performance_train['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Optimal Support Vector Machine Model Performance on Train Data: ')
display(optimal_support_vector_machine_performance_train)

Optimal Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.947368	optimal_support_vector_machine	train
1	Precision	0.960000	optimal_support_vector_machine	train
2	Recall	0.827586	optimal_support_vector_machine	train
3	F1	0.888889	optimal_support_vector_machine	train
4	AUROC	0.907911	optimal_support_vector_machine	train

##################################
# Evaluating the optimal Support Vector Machine model
# on the test set
##################################
optimal_support_vector_machine_y_hat_test = optimal_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
optimal_support_vector_machine_performance_test = model_performance_evaluation(y_test, optimal_support_vector_machine_y_hat_test)
optimal_support_vector_machine_performance_test['model'] = ['optimal_support_vector_machine'] * 5
optimal_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Optimal Support Vector Machine Model Performance on Test Data: ')
display(optimal_support_vector_machine_performance_test)

Optimal Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.857143	optimal_support_vector_machine	test
1	Precision	0.857143	optimal_support_vector_machine	test
2	Recall	0.500000	optimal_support_vector_machine	test
3	F1	0.631579	optimal_support_vector_machine	test
4	AUROC	0.736486	optimal_support_vector_machine	test

1.7. Model Development With Class Weights

Class weights are used to assign different levels of importance to different classes when the distribution of instances across different classes in a classification problem is not equal. By assigning higher weights to the minority class, the model is encouraged to give more attention to correctly predicting instances from the minority class. Class weights are incorporated into the loss function during training. The loss for each instance is multiplied by its corresponding class weight. This means that misclassifying an instance from the minority class will have a greater impact on the overall loss than misclassifying an instance from the majority class. The use of class weights helps balance the influence of each class during training, mitigating the impact of class imbalance. It provides a way to focus the learning process on the classes that are underrepresented in the training data.

1.7.1 Premodelling Data Description

##################################
# Consolidating relevant numeric columns
# and encoded categorical columns
# after hypothesis testing
##################################
cancer_rate_premodelling = cancer_rate_preprocessed_all.drop(['AGRLND','POPDEN','GHGEMI','POPGRO','FORARE','HDICAT_H','HDICAT_M','HDICAT_L'], axis=1)

##################################
# Performing a general exploration of the filtered dataset
##################################
print('Dataset Dimensions: ')
display(cancer_rate_premodelling.shape)

Dataset Dimensions: 

(163, 9)

##################################
# Listing the column names and data types
##################################
print('Column Names and Data Types:')
display(cancer_rate_premodelling.dtypes)

Column Names and Data Types:

URBPOP        float64
LIFEXP        float64
TUBINC        float64
DTHCMD        float64
CO2EMI        float64
GDPCAP        float64
EPISCO        float64
CANRAT       category
HDICAT_VH       uint8
dtype: object

##################################
# Gathering the pairplot for all variables
##################################
sns.pairplot(cancer_rate_premodelling, kind='reg')
plt.show()

##################################
# Separating the target 
# and predictor columns
##################################
X = cancer_rate_premodelling.drop('CANRAT', axis = 1)
y = cancer_rate_premodelling['CANRAT'].cat.codes

##################################
# Formulating the train and test data
# using a 70-30 ratio
##################################
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.3, random_state= 88888888, stratify=y)

##################################
# Performing a general exploration of the train dataset
##################################
print('Dataset Dimensions: ')
display(X_train.shape)

Dataset Dimensions: 

(114, 8)

##################################
# Validating the class distribution of the train dataset
##################################
y_train.value_counts(normalize = True)

0    0.745614
1    0.254386
dtype: float64

##################################
# Performing a general exploration of the test dataset
##################################
print('Dataset Dimensions: ')
display(X_test.shape)

Dataset Dimensions: 

(49, 8)

##################################
# Validating the class distribution of the test dataset
##################################
y_test.value_counts(normalize = True)

0    0.755102
1    0.244898
dtype: float64

##################################
# Defining a function to compute
# model performance
##################################
def model_performance_evaluation(y_true, y_pred):
    metric_name = ['Accuracy','Precision','Recall','F1','AUROC']
    metric_value = [accuracy_score(y_true, y_pred),
                   precision_score(y_true, y_pred),
                   recall_score(y_true, y_pred),
                   f1_score(y_true, y_pred),
                   roc_auc_score(y_true, y_pred)]    
    metric_summary = pd.DataFrame(zip(metric_name, metric_value),
                                  columns=['metric_name','metric_value']) 
    return(metric_summary)

1.7.2 Logistic Regression

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

##################################
# Creating an instance of the 
# Logistic Regression model
##################################
logistic_regression = LogisticRegression()

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'penalty': ['l1', 'l2'],
    'solver': ['liblinear','saga'],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Logistic Regression model
##################################
weighted_logistic_regression = GridSearchCV(estimator = logistic_regression, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Logistic Regression model
##################################
weighted_logistic_regression.fit(X_train, y_train)

##################################
# Determining the weighted hyperparameter
# for the Logistic Regression model
##################################
weighted_logistic_regression.best_score_ 
weighted_logistic_regression.best_params_

{'C': 1.0,
 'class_weight': {0: 0.25, 1: 0.75},
 'penalty': 'l2',
 'random_state': 88888888,
 'solver': 'liblinear'}

##################################
# Evaluating the weighted logistic regression model
# on the train set
##################################
weighted_logistic_regression_y_hat_train = weighted_logistic_regression.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_train = model_performance_evaluation(y_train, weighted_logistic_regression_y_hat_train)
weighted_logistic_regression_performance_train['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_train['set'] = ['train'] * 5
print('Weighted Logistic Regression Model Performance on Train Data: ')
display(weighted_logistic_regression_performance_train)

Weighted Logistic Regression Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.894737	weighted_logistic_regression	train
1	Precision	0.707317	weighted_logistic_regression	train
2	Recall	1.000000	weighted_logistic_regression	train
3	F1	0.828571	weighted_logistic_regression	train
4	AUROC	0.929412	weighted_logistic_regression	train

##################################
# Evaluating the weighted logistic regression model
# on the test set
##################################
weighted_logistic_regression_y_hat_test = weighted_logistic_regression.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_logistic_regression_performance_test = model_performance_evaluation(y_test, weighted_logistic_regression_y_hat_test)
weighted_logistic_regression_performance_test['model'] = ['weighted_logistic_regression'] * 5
weighted_logistic_regression_performance_test['set'] = ['test'] * 5
print('Weighted Logistic Regression Model Performance on Test Data: ')
display(weighted_logistic_regression_performance_test)

Weighted Logistic Regression Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.938776	weighted_logistic_regression	test
1	Precision	0.846154	weighted_logistic_regression	test
2	Recall	0.916667	weighted_logistic_regression	test
3	F1	0.880000	weighted_logistic_regression	test
4	AUROC	0.931306	weighted_logistic_regression	test

1.7.3 Decision Trees

Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)

##################################
# Creating an instance of the 
# Decision Tree model
##################################
decision_tree = DecisionTreeClassifier()

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Decision Tree model
##################################
weighted_decision_tree = GridSearchCV(estimator = decision_tree, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Decision Tree model
##################################
weighted_decision_tree.fit(X_train, y_train)

##################################
# Determining the weighted hyperparameter
# for the Decision Tree model
##################################
weighted_decision_tree.best_score_ 
weighted_decision_tree.best_params_

{'class_weight': {0: 0.25, 1: 0.75},
 'criterion': 'gini',
 'max_depth': 3,
 'min_samples_leaf': 3,
 'random_state': 88888888}

##################################
# Evaluating the weighted decision tree model
# on the train set
##################################
weighted_decision_tree_y_hat_train = weighted_decision_tree.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_train = model_performance_evaluation(y_train, weighted_decision_tree_y_hat_train)
weighted_decision_tree_performance_train['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_train['set'] = ['train'] * 5
print('Weighted Decision Tree Model Performance on Train Data: ')
display(weighted_decision_tree_performance_train)

Weighted Decision Tree Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.956140	weighted_decision_tree	train
1	Precision	0.852941	weighted_decision_tree	train
2	Recall	1.000000	weighted_decision_tree	train
3	F1	0.920635	weighted_decision_tree	train
4	AUROC	0.970588	weighted_decision_tree	train

##################################
# Evaluating the weighted decision tree model
# on the test set
##################################
weighted_decision_tree_y_hat_test = weighted_decision_tree.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_decision_tree_performance_test = model_performance_evaluation(y_test, weighted_decision_tree_y_hat_test)
weighted_decision_tree_performance_test['model'] = ['weighted_decision_tree'] * 5
weighted_decision_tree_performance_test['set'] = ['test'] * 5
print('Weighted Decision Tree Model Performance on Test Data: ')
display(weighted_decision_tree_performance_test)

Weighted Decision Tree Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.897959	weighted_decision_tree	test
1	Precision	0.769231	weighted_decision_tree	test
2	Recall	0.833333	weighted_decision_tree	test
3	F1	0.800000	weighted_decision_tree	test
4	AUROC	0.876126	weighted_decision_tree	test

1.7.4 Random Forest

Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.

##################################
# Creating an instance of the 
# Random Forest model
##################################
random_forest = RandomForestClassifier()

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
hyperparameter_grid = {
    'criterion': ['gini','entropy','log_loss'],
    'max_depth': [3,5,7],
    'min_samples_leaf': [3,5,10],
    'n_estimators': [3,5,7],
    'max_features':['sqrt', 'log2'],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Random Forest model
##################################
weighted_random_forest = GridSearchCV(estimator = random_forest, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Random Forest model
##################################
weighted_random_forest.fit(X_train, y_train)

##################################
# Determining the weighted hyperparameter
# for the Random Forest model
##################################
weighted_random_forest.best_score_ 
weighted_random_forest.best_params_

{'class_weight': {0: 0.25, 1: 0.75},
 'criterion': 'entropy',
 'max_depth': 3,
 'max_features': 'sqrt',
 'min_samples_leaf': 5,
 'n_estimators': 3,
 'random_state': 88888888}

##################################
# Evaluating the weighted Random Forest model
# on the train set
##################################
weighted_random_forest_y_hat_train = weighted_random_forest.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_train = model_performance_evaluation(y_train, weighted_random_forest_y_hat_train)
weighted_random_forest_performance_train['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_train['set'] = ['train'] * 5
print('Weighted Random Forest Model Performance on Train Data: ')
display(weighted_random_forest_performance_train)

Weighted Random Forest Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.929825	weighted_random_forest	train
1	Precision	0.800000	weighted_random_forest	train
2	Recall	0.965517	weighted_random_forest	train
3	F1	0.875000	weighted_random_forest	train
4	AUROC	0.941582	weighted_random_forest	train

##################################
# Evaluating the weighted Random Forest model
# on the test set
##################################
weighted_random_forest_y_hat_test = weighted_random_forest.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_random_forest_performance_test = model_performance_evaluation(y_test, weighted_random_forest_y_hat_test)
weighted_random_forest_performance_test['model'] = ['weighted_random_forest'] * 5
weighted_random_forest_performance_test['set'] = ['test'] * 5
print('Weighted Random Forest Model Performance on Test Data: ')
display(weighted_random_forest_performance_test)

Weighted Random Forest Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.938776	weighted_random_forest	test
1	Precision	0.909091	weighted_random_forest	test
2	Recall	0.833333	weighted_random_forest	test
3	F1	0.869565	weighted_random_forest	test
4	AUROC	0.903153	weighted_random_forest	test

1.7.5 Support Vector Machine

Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.

##################################
# Creating an instance of the 
# Support Vector Machine model
##################################
support_vector_machine = SVC()

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
hyperparameter_grid = {
    'C': [1.0],
    'kernel': ['linear', 'poly', 'rbf', 'sigmoid'],
    'class_weight': [{0:0.25, 1:0.75}],
    'random_state': [88888888]}

##################################
# Defining the hyperparameters for the
# Support Vector Machine model
##################################
weighted_support_vector_machine = GridSearchCV(estimator = support_vector_machine, 
                                           param_grid = hyperparameter_grid,
                                           n_jobs = -1,
                                           scoring='f1')

##################################
# Fitting the weighted Support Vector Machine model
##################################
weighted_support_vector_machine.fit(X_train, y_train)

##################################
# Determining the weighted hyperparameter
# for the Support Vector Machine model
##################################
weighted_support_vector_machine.best_score_ 
weighted_support_vector_machine.best_params_

{'C': 1.0,
 'class_weight': {0: 0.25, 1: 0.75},
 'kernel': 'poly',
 'random_state': 88888888}

##################################
# Evaluating the weighted Support Vector Machine model
# on the train set
##################################
weighted_support_vector_machine_y_hat_train = weighted_support_vector_machine.predict(X_train)

##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_train = model_performance_evaluation(y_train, weighted_support_vector_machine_y_hat_train)
weighted_support_vector_machine_performance_train['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_train['set'] = ['train'] * 5
print('Weighted Support Vector Machine Model Performance on Train Data: ')
display(weighted_support_vector_machine_performance_train)

Weighted Support Vector Machine Model Performance on Train Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.964912	weighted_support_vector_machine	train
1	Precision	0.962963	weighted_support_vector_machine	train
2	Recall	0.896552	weighted_support_vector_machine	train
3	F1	0.928571	weighted_support_vector_machine	train
4	AUROC	0.942394	weighted_support_vector_machine	train

##################################
# Evaluating the weighted Support Vector Machine model
# on the test set
##################################
weighted_support_vector_machine_y_hat_test = weighted_support_vector_machine.predict(X_test)

##################################
# Gathering the model evaluation metrics
##################################
weighted_support_vector_machine_performance_test = model_performance_evaluation(y_test, weighted_support_vector_machine_y_hat_test)
weighted_support_vector_machine_performance_test['model'] = ['weighted_support_vector_machine'] * 5
weighted_support_vector_machine_performance_test['set'] = ['test'] * 5
print('Weighted Support Vector Machine Model Performance on Test Data: ')
display(weighted_support_vector_machine_performance_test)

Weighted Support Vector Machine Model Performance on Test Data: 

	metric_name	metric_value	model	set
0	Accuracy	0.877551	weighted_support_vector_machine	test
1	Precision	0.875000	weighted_support_vector_machine	test
2	Recall	0.583333	weighted_support_vector_machine	test
3	F1	0.700000	weighted_support_vector_machine	test
4	AUROC	0.778153	weighted_support_vector_machine	test

1.8. Model Development With SMOTE Upsampling

1.8.1 Premodelling Data Description

1.8.2 Logistic Regression

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

1.8.3 Decision Trees

Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)

1.8.4 Random Forest

Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.

1.8.5 Support Vector Machine

Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.

1.9. Model Development With CNN Downsampling

1.9.1 Premodelling Data Description

1.9.2 Logistic Regression

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

1.9.3 Decision Trees

Decision trees create a model that predicts the class label of a sample based on input features. A decision tree consists of nodes that represent decisions or choices, edges which connect nodes and represent the possible outcomes of a decision and leaf (or terminal) nodes which represent the final decision or the predicted class label. The decision-making process involves feature selection (at each internal node, the algorithm decides which feature to split on based on a certain criterion including gini impurity or entropy), splitting criteria (the splitting criteria aim to find the feature and its corresponding threshold that best separates the data into different classes. The goal is to increase homogeneity within each resulting subset), recursive splitting (the process of feature selection and splitting continues recursively, creating a tree structure. The dataset is partitioned at each internal node based on the chosen feature, and the process repeats for each subset) and stopping criteria (the recursion stops when a certain condition is met, known as a stopping criterion. Common stopping criteria include a maximum depth for the tree, a minimum number of samples required to split a node, or a minimum number of samples in a leaf node.)

1.9.4 Random Forest

Random Forest is an ensemble learning method made up of a large set of small decision trees called estimators, with each producing its own prediction. The random forest model aggregates the predictions of the estimators to produce a more accurate prediction. The algorithm involves bootstrap aggregating (where smaller subsets of the training data are repeatedly subsampled with replacement), random subspacing (where a subset of features are sampled and used to train each individual estimator), estimator training (where unpruned decision trees are formulated for each estimator) and inference by aggregating the predictions of all estimators.

1.9.5 Support Vector Machine

Support Vector Machine plots each observation in an N-dimensional space corresponding to the number of features in the data set and finds a hyperplane that maximally separates the different classes by a maximally large margin (which is defined as the distance between the hyperplane and the closest data points from each class). The algorithm applies kernel transformation by mapping non-linearly separable data using the similarities between the points in a high-dimensional feature space for improved discrimination.

1.10. Model Development With Stacking Ensemble Learning

1.10.1 Premodelling Data Description

1.10.2 Logistic Regression

Logistic Regression models the relationship between the probability of an event (among two outcome levels) by having the log-odds of the event be a linear combination of a set of predictors weighted by their respective parameter estimates. The parameters are estimated via maximum likelihood estimation by testing different values through multiple iterations to optimize for the best fit of log odds. All of these iterations produce the log likelihood function, and logistic regression seeks to maximize this function to find the best parameter estimates. Given the optimal parameters, the conditional probabilities for each observation can be calculated, logged, and summed together to yield a predicted probability.

2. Summary

3. References

• [Book] Data Preparation for Machine Learning: Data Cleaning, Feature Selection, and Data Transforms in Python by Jason Brownlee
• [Book] Feature Engineering and Selection: A Practical Approach for Predictive Models by Max Kuhn and Kjell Johnson
• [Book] Feature Engineering for Machine Learning by Alice Zheng and Amanda Casari
• [Book] Applied Predictive Modeling by Max Kuhn and Kjell Johnson
• [Book] Data Mining: Practical Machine Learning Tools and Techniques by Ian Witten, Eibe Frank, Mark Hall and Christopher Pal
• [Book] Data Cleaning by Ihab Ilyas and Xu Chu
• [Book] Data Wrangling with Python by Jacqueline Kazil and Katharine Jarmul
• [Book] Regression Modeling Strategies by Frank Harrell
• [Python Library API] NumPy by NumPy Team
• [Python Library API] pandas by Pandas Team
• [Python Library API] seaborn by Seaborn Team
• [Python Library API] matplotlib.pyplot by MatPlotLib Team
• [Python Library API] itertools by Python Team
• [Python Library API] operator by Python Team
• [Python Library API] sklearn.experimental by Scikit-Learn Team
• [Python Library API] sklearn.impute by Scikit-Learn Team
• [Python Library API] sklearn.linear_model by Scikit-Learn Team
• [Python Library API] sklearn.preprocessing by Scikit-Learn Team
• [Python Library API] scipy by SciPy Team
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• [Article] Easy Guide To Data Preprocessing In Python by Ahmad Anis (KDNuggets)
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• [Article] Ordinal and One-Hot Encodings for Categorical Data by Jason Brownlee (Machine Learning Mastery)
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• [Publication] Data Quality for Machine Learning Tasks by Nitin Gupta, Shashank Mujumdar, Hima Patel, Satoshi Masuda, Naveen Panwar, Sambaran Bandyopadhyay, Sameep Mehta, Shanmukha Guttula, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’21: Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining)
• [Publication] Overview and Importance of Data Quality for Machine Learning Tasks by Abhinav Jain, Hima Patel, Lokesh Nagalapatti, Nitin Gupta, Sameep Mehta, Shanmukha Guttula, Shashank Mujumdar, Shazia Afzal, Ruhi Sharma Mittal and Vitobha Munigala (KDD ’20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining)
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